Problem 19 - Calculus Concepts

Question

ATM Surcharges \(99.2 \%\) of ATMs levy a surcharge on users who are not account holders. The amount of the surcharge for non-account holders can be modeled as $$ s(t)=0.72\left(1.081^{t}\right) \text { dollars } $$ a. Calculate the average rate of change in the amount of the surcharge for non-account holders between 1998 and 2008\. Write the result in a sentence of interpretation. b. Calculate the change and the percentage change in the amount of the surcharge for non-account holders between 1998 and 2008 . where \(t\) is the number of years since \(1995,\) data from \(3 \leq t \leq 13\)

Step-by-Step Solution

Answer

The surcharge increased by approximately \$0.79 or 86.5% from 1998 to 2008, with an average rate of \$0.079 per year.

Step 1: Identify the Years in Terms of t

Since \( t \) is the number of years since 1995, for 1998, we have \( t = 3 \) (because \( 1998 - 1995 = 3 \)), and for 2008, we have \( t = 13 \) (because \( 2008 - 1995 = 13 \)).

Step 2: Calculate the Surcharge for t = 3

Substitute \( t = 3 \) into the surcharge formula: \( s(3) = 0.72 \times 1.081^3 \). Calculate this to find the surcharge in 1998.

Step 3: Evaluate s(3)

Calculate \( s(3) \):\[ s(3) = 0.72 \times (1.081)^3 \approx 0.72 \times 1.2644 \approx 0.9104 \]Thus, the surcharge in 1998 was approximately \$0.91.

Step 4: Calculate the Surcharge for t = 13

Substitute \( t = 13 \) into the surcharge formula: \( s(13) = 0.72 \times 1.081^{13} \). Calculate this to find the surcharge in 2008.

Step 5: Evaluate s(13)

Calculate \( s(13) \):\[ s(13) = 0.72 \times (1.081)^{13} \approx 0.72 \times 2.3579 \approx 1.6977 \]Thus, the surcharge in 2008 was approximately \$1.70.

Step 6: Calculate Average Rate of Change

The average rate of change between 1998 and 2008 is the difference in surcharge values divided by the change in time:\[ \frac{s(13) - s(3)}{13 - 3} = \frac{1.6977 - 0.9104}{10} \approx \frac{0.7873}{10} \approx 0.07873 \text{ dollars per year} \]This means the surcharge increased by approximately \$0.079 per year on average.

Step 7: Calculate Total Change and Percentage Change

The total change in surcharge from 1998 to 2008 is:\[ s(13) - s(3) = 1.6977 - 0.9104 = 0.7873 \text{ dollars} \]The percentage change is:\[ \left( \frac{0.7873}{0.9104} \right) \times 100\% \approx 86.5\% \]The surcharge increased by approximately \$0.79 or 86.5% over the period.

Key Concepts

Understanding the Average Rate of Change

The average rate of change is a fundamental concept in calculus that describes how a quantity changes on average over a specific interval. To put it simply, it's like finding the "slope" of the line connecting two points on a graph. In the context of our problem, we're looking at how the ATM surcharge for non-account holders changed between 1998 and 2008.In technical terms, it's calculated as:- The change in the value of the function: here, it’s the surcharge amount, from the beginning of the period to the end;- Divided by the period: in our case, the 10 years between 1998 and 2008.Using the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \] where \( s(t_1) \) and \( s(t_2) \) are the surcharges at the beginning and end of the period, we find that the charge increased at an average rate of \\(0.079 per year. This means each year, on average, non-account holders faced a \\)0.079 increase in the surcharge.

Exploring Exponential Growth

Exponential growth refers to an increase that becomes quicker as the thing that is growing becomes larger. It’s different from a linear growth, which increases at a constant rate. In exponential growth, each quantity grows by a percentage of its current size, so it keeps getting bigger and bigger. This is a common pattern found in many natural phenomena and financial calculations. In our surcharge model, the increase is depicted as an exponential function.The formula given for the surcharge is: \[ s(t) = 0.72 \times 1.081^t \] Here, \(1.081\) is the base of the exponential function, which means the surcharge grows by approximately 8.1% each year. The surcharge doesn’t just increase by a flat dollar amount each year. Instead, the added amount for each year is a bit more than the previous year’s because it’s based on the existing amount.Exponential growth is powerful because even if changes seem small at first, they compound to become substantial over time. The surcharge grew from about \\(0.91 in 1998 to \\)1.70 in 2008, illustrating this principle. This kind of growth underscores why small percentages can lead to significant changes over longer periods.

Calculating Percentage Change

Percentage change is a useful way to describe how much a quantity has increased or decreased relative to its original value. It's often more intuitive than absolute changes, especially in financial contexts where changes might seem small.The formula for percentage change is: \[ \text{Percentage Change} = \left( \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \right) \times 100\% \] In our example, we calculate how much the ATM surcharge changed from 1998 to 2008.- The surcharge in 1998 was approximately \\(0.91.- The surcharge in 2008 was approximately \\)1.70.The total dollar change is \(1.70 - 0.91 = 0.79\), and expressing this as a percentage of the 1998 value gives us: \[ \left( \frac{0.79}{0.91} \right) \times 100\% \approx 86.5\% \] This tells us that the surcharge increased by 86.5% over the decade. Understanding percentage change helps us grasp the relative significance of the increase, providing a clearer view of how impactful the change is compared to its initial size.